$\begingroup$Something along these lines might work. $a^2+a+1$, $b^2+b+1$, $(a+b)^2-2(a+b)+4=(a+b-1)^2+3$ are all increasing functions with respect to $a$ and $b$. The infimum of the set of values the LHS can take is therefore $\lim_{a\rightarrow \infty }\lim_{b\rightarrow \infty }\left ( \frac{1}{\sqrt{a^{2}+a+1}}+\frac{1}{\sqrt{b^{2}+b+1}} + \frac{4}{\sqrt{\left ( a+b \right )^{2}-2\left ( a+b \right )+4}} \right ) = 0$. Less than $4$, so simply choose $a,b$ large enough. This strengthens the result to the RHS being any non-zero positive number.$\endgroup$
– LPenguinSep 23 '18 at 1:01